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A217670
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G.f.: Sum_{n>=0} x^n/(1 + x^n)^n.
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7
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1, 1, 0, 2, -2, 2, 0, 2, -8, 8, 0, 2, -12, 2, 0, 32, -36, 2, 0, 2, -20, 58, 0, 2, -136, 72, 0, 92, -28, 2, 0, 2, -272, 134, 0, 422, -288, 2, 0, 184, -480, 2, 0, 2, -44, 1232, 0, 2, -2360, 926, 0, 308, -52, 2, 0, 2004, -1176, 382, 0, 2, -4064, 2, 0, 6470, -5128, 3642
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OFFSET
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0,4
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LINKS
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FORMULA
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a(4*n+2) = 0 for n>=0.
a(n) = Sum_{d|n} (-1)^(d-1) * binomial(d+n/d-2, d-1) for n > 0.
If p is prime, a(p) = 1 + (-1)^(p-1). (End)
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^3 - 2*x^4 + 2*x^5 + 2*x^7 - 8*x^8 + 8*x^9 +...
where
A(x) = 1 + x/(1+x) + x^2/(1+x^2)^2 + x^3/(1+x^3)^3 + x^4/(1+x^4)^4 + x^5/(1+x^5)^5 +...
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MATHEMATICA
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terms = 100; Sum[x^n/(1 + x^n)^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)
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PROG
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(PARI) {a(n)=polcoeff(sum(m=0, n, x^m/(1+x^m +x*O(x^n))^m), n)}
for(n=0, 100, print1(a(n), ", "))
(PARI) a(n) = if(n==0, 1, sumdiv(n, d, (-1)^(d-1)*binomial(d+n/d-2, d-1))); \\ Seiichi Manyama, Apr 23 2021
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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